Thursday, 25 August 2016

calculus - how to strictly prove $sin x



$$\sin xIn most textbooks, to prove this inequality is based on geometry illustration (draw a circle, compare arc length and chord ), but I think that strict proof should be based on analysis reasoning without geometry illustration. Who can prove it? Thank you very much.






ps:





  1. By differentiation, monotonicity and Taylor formula, all are wrong, because (sinx)=cosx must use limx0sinxx=1, and this formula must use sinx<x. This is vicious circle.


  2. If we use Taylor series of sinx to define sinx, strictly prove $\sin x



Answer



We can define sinx as power series. Applying the knowledge of power series, obtain the derivative of sinx, and then we will easy prove the inequality. Concluding geometry of sinx, please refer to this.


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